maths problem solving open ended questions

• ELEMENTARY TEACHING , MATH

Open Ended Math Questions and Problems for Elementary Students

Does your current math instruction involve only situations where there is one answer? Are students expected to solve problems following rigid procedures that do not require critical or creative thinking? These components are important; however, it’s time to take your math instruction to the next level! The answer: open ended math questions!

Open ended math questions, also known as open ended math problems, help learners grow into true mathematicians who use diverse problem solving strategies to explore mathematical situations where there isn’t necessarily one “right” answer. It equips them with the critical thinking skills they need to solve real world problems in the twenty-first century.

This blog post will answer the following questions:

• What is an open ended math question?
• What are the differences between open-ended and closed-ended problems in math?
• Why should I implement open ended questions in my classroom?
• What are the disadvantages of using open-ended math problems?
• How do I implement open ended math questions in my classroom?
• How do I create open ended math questions?
• What are some examples of open ended math problems?
• How do I grade open ended math tasks?

What is an Open Ended Math Question?

An open ended math question (which is known as an open ended math problem or open ended math task) is a real world math situation presented to students in a word problem format where there is more than one solution, approach, and representation. This instructional strategy is more than reciting a fact or repeating a procedure. It requires students to apply what they have learned while using their problem solving, reasoning, critical thinking, and communication skills to solve a given problem.

This strategy naturally allows for differentiation because of its open-ended nature. In addition, it is a valuable formative assessment tool that allows teachers to assess accuracy in computation and abilities to think of and flexibly apply more than one strategy. In addition to the teacher being able to learn about their students from this tool, the students can thoughtfully extend their learning and reflect on their own thinking through whole group discussions or partner talks.

What are the Differences Between Open-Ended and Closed-Ended Problems in Math?

The major difference between open-ended math problems and closed-ended math problems is that close-ended ones have one answer and open-ended ones have more than one answer. This simple difference creates a very different learning experience for elementary students when they work on solving the problem.

What are the Advantages and Disadvantages of Open Ended Math Problems?

Advantages of open ended math problems.

There are many benefits to using open ended math questions in your classroom. This list of advantages of open ended questions will help you understand their ability to transform your math block! Here are 8 advantages to using open ended math tasks:

• Provides valuable and specific information to the teacher about student understanding and application of learning
• Allows the teacher to assess accuracy in computation and abilities to think of and flexibly apply more than one strategy
• Permits the teacher to see flexibility in student thinking
• Gives students the opportunity to practice and fine tune their problems solving, reasoning, critical thinking, and communication skills
• Creates opportunities for real-world application of math
• Empowers students to extend their learning and reflect on their thinking
• Fosters creativity, collaboration, and engagement in students
• Facilitates a differentiated learning experience where all students can access the task

Disadvantages of Open Ended Math Problems

Although there are tons of benefits to using open ended math problems in your classroom, it is important to note that there are some disadvantages. Here are 3 disadvantages using open ended math tasks:

• Increases time in collecting data
• Provides a higher complexity of data
• Requires the implementation and practice of routines

3 Ways to Implement Open Ended Problems During Your Math Block

Here are 3 ways you can implement open ended problems in your elementary classroom:

• Start a lesson with an open-ended math problems for students to solve independently. Invite them to share their work and reasoning with a partner. Ask a few students to share their ideas with the whole class.
• Use the open-ended math problems for fast finishers . If a student or a group of students tend to finish independent work before the rest of the class, invite them to work on an open-ended math problem.
• Utilize open-ended math problems as a center during math workshop . You will not have to worry about students finishing that math center before it is time to switch to the next center.

3 Ways to Write Open Ended Math Questions with Examples

Here are 3 ways to create open ended math questions accompanied with easy-to-understand open ended math problems examples:

• Start with a Closed-Ended Question. For example, a closed-ended question could be: What is the sum of 10 plus 10? The related open-ended question would be: The sum is 20. What could the addends be? There are an infinite number of responses because students could use negative numbers.
• Ask Students to Explain, Prove, or Justify their Thinking. An example of this is, “Prove 5 + 6 = 11.” One possible student response could be that they know the sum is 11 because of the doubles + 1 rule. Another student may take out counters, while another draws a picture.
• Invite Students to Compare 2 Concepts. For example, ask students to identify the similarities and differences between 2D and 3D shapes. Some possible responses for similarities are that they are both geometry concepts and classifications of shapes. A difference they could say is that 2D shapes are flat, while a 3D shape is solid.

How do you Grade Open-Ended Math Questions?

Grading open-ended math tasks is not as clear cut as closed-ended questions. If you are using the task as a formative assessment for your own planning purposes, then you have flexibility on how you choose to evaluate students’ work. However, I recommend you use a rubric if you plan to use it as a summative assessment. Remember to share the rubric with students so that the expectations are clear.

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We would love for you to try these open ended question resources with your students. They offer students daily opportunities to practice solving open ended problems. You can download worksheets specific to your grade level (along with lots of other math freebies) in our free printable math resources bundle using this link: free printable math worksheets for elementary teachers .

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Opening Minds with Open-Ended Math Problems in the Primary Classroom

by Model Teaching | December 21, 2018.

Wait, so, what is the RIGHT answer?” “Sarah got a different answer than I did…how can we BOTH be right?” You will most likely hear all kinds of responses like this when you start to incorporate open-ended math activities into your classroom. At first, they’ll probably make your students look at you as if you have two heads. But, these kinds of reactions will begin to subside once your students have been exposed to the idea that there are many ways to solve problems, even math problems! Encouraging this kind of “endless possibility” thinking is an effective way to teach your students to challenge themselves and think outside of the “normal” problem solving thinking.

What are open-ended math problems?

Open-ended math problems are problems that have more than one possible answer. These problems might present an end result and then ask students to work backward to figure out how that end result might have been achieved or they might ask students to compare two concepts that can be compared in a variety of different ways. But whatever way they are presented, the purpose of open-ended math problems is always to encourage students to use higher order thinking skills to solve problems and understand that some problems can be solved in many ways, with many outcomes.

Examples of Open-ended Math Problems

If you teach pre-k or kindergarten, an open-ended math problem might be: “You have 2 shapes that have a different amount of sides. What 2 shapes could you have? Show and name the shapes.” You would provide them with crayons, paper, pattern blocks, or whatever other manipulative they might be used to using when discussing shapes and students would use these manipulatives to come up with as many answers as they can. Your little ones may answer with a variety of answers based on their current skill level. You may get answers like “triangle and square”, “hexagon and parallelogram”, or “a circle is a shape” depending on what each student knows about shapes. This is a great way to reinforce what students already know and to quickly assess where they are in their knowledge.

If you teach first grade, an open-ended math problem might be: “I’m thinking of the number 8. What two numbers work could work together to make the number 8?” Again, you would provide them with manipulatives they normally use for composing and decomposing numbers, like counters, small erasers, counting bears, unifix cubes, or even playdoh balls. The extra bonus about this kind of problem is that it’s extremely easy for students to show their math skills. Some might use addition, others will use subtraction, and you may even run into a kiddo or two who can use multiplication to find the number. However students choose to explore all the possibilities for answers, be sure to give them a few options for how to show their thinking. This might include simply writing equations, drawing pictures with the equations, or even building the number with a manipulative and then taking a picture of it with an iPad.

As students get older and move onto more abstract thinking in second and third grade, you might incorporate more word problems like: “The difference between the temperature on Monday and Tuesday was 13 degrees. What could the temperature have been on each day? Find and explain at least 5 different answers.” Or “Penelope sees 37 children playing in a corn maze. If those children split into four groups, how many children could be in each group? Find and explain at least 5 different answers.” As always, be sure to provide students with manipulatives, paper and pencils, dry erase markers and whiteboards, or whatever you normally use to help them solve problems and then let them go to work! By presenting these kinds of word problems, you’ll expose students to a variety of math concepts (such as division in this example) just by allowing them to think about how to solve the problem on their own. Then, when these concepts are formally introduced, they will hopefully feel more familiar to some students.

Why should I use open-ended math problems with my students?

There are many benefits to incorporating these kinds of problems into your students’ daily routine, but here are a few of the most obvious and effective ones:

• Open-ended problems encourage higher order thinking skills. Students will not only be “recognizing”, “identifying”, or “describing” their thinking; they’ll be “justifying”, “defending”, and “evaluating” their problem solving skills and how they arrived at their answers.
• Open-ended problems build confidence in your students. Once students recognize that there are many possibilities for correct answers and thinking, they begin to participate more readily because they can bring to the table. Students who normally struggle with math might solve the problem on a very basic level, using a basic strategy, but they’ll be correct! And your advanced students can solve it on their advanced level and be just as correct as the student who struggles. Simply knowing that the way that they specifically thought and solved the problem was considered correct builds confidence for students.
• Open-ended problems are engaging! Students are immediately engaged in these kinds of problems because they recognize that there are so many different ways to solve it. Whether students are working in small groups or independently, there is possibility for so many different ideas and answers to be correct that everyone wants in on it. This engagement, in turn, encourages collaboration among students and soon, they’re sharing their thinking and learning from each other to solve problems.
• Open-ended problems encourage creativity. Students are capable of using so many strategies that they’ve learned over the years to solve problems and, given the space and time, they can even come up with some of their own strategies for solving problems. Open-ended problems give students permission to be creative in their thinking and problem solving.
• Open-ended problems make it easy for teachers to see what levels students are working at. Simply by walking around the room while students are working to solve an open-ended math problem, you’ll be able to informally assess what kind of level they are independently working on. This can be extremely beneficial as you are collecting data, forming groups, or simply getting a feel for what kind of skills each student is working with.

For more information about the benefits of using open-ended math problems, read:

https://nzmaths.co.nz/benefits-problem-solving

How do I incorporate open-ended math problems into my math instructional time?

Some of the simplest ways to incorporate open-ended math problems into your math instructional time is to include them in math stations, use them in small groups, and use them as a warm up.

• Math Stations:  You can implement open-ended problems into your math stations a number of ways, including thinking mats, task cards, or interactive math journals. The simplest way to implement them into your math stations is by using task cards. Task cards are pre-made cards that you can create or purchase to cut and laminate for students to use repeatedly. Task cards usually include words, pictures, diagrams, or a combination of all to present a problem to students. To use task cards in a math station, simply create or purchase the cards you want with open-ended word problems or picture problems. Then, simply print them out and cut/laminate them to make them durable and easily reused. (TEACHER TIP: Most dry erase markers wipe off of lamination pretty easily if it’s wiped off within a reasonable amount of time. Your students may want to mark the important parts of the problem on the actual task card with dry erase marker if you want them to. Just wipe if off after use!) I would suggest storing cards in a labeled plastic container or ziplock bag to keep them organized. It is suggested that you always allow students to use manipulatives as needed, as this can help students feel allowed to express their creative problem-solving thoughts. So, be sure your task card station provides anything students might use to solve problems in their own way: whiteboards, markers, papers, crayons, counters, manipulatives, thinking mats, laminated task cards, etc. For example, if you give students a task card with this problem on it: “Marcy finds 47 apples on the ground. What 3 addends could create this sum? Find and explain at least 5 answers.”, I would provide them with small apple erasers or counters, a whiteboard and dry erase marker, and an iPad to take picture evidence of their five (or more!) answers when they’re finished.  Please refer to pages 10-16 in the resource provided to you below this article for some sample open-ended word problem task cards that you can use with your students immediately!
• Small Groups: To implement open-ended problems in your small groups, using thinking mats, manipulatives, and prepared open-ended problems is a great way to ease students into working on open-ended problems independently. This is a great way to model your own thinking and problem-solving to allow students to see how they can begin their own ways of solving the problems. Take a moment to download and look at the thinking mat activity in the downloadable resource below. You can incorporate these mats into your small group activity by providing each student with a laminated copy of the mat you want to use and manipulatives for them to work with to follow the mat’s directions. For example, the thinking mat that says “Make patterns out of these shapes and name them.” would be an excellent open-ended activity for a group of kindergartners who are working on shapes OR patterns. Give each student a few of each of the pattern blocks shown on the mat and a dry erase marker. Explain and model how YOU would complete the activity by creating a pattern with the pattern blocks, tracing the blocks or drawing your pattern, and then naming it with letters (ie.: rhombus, rhombus, circle would be named an AAB pattern). This will give your students an idea of what’s expected and their little brains can get started coming up with their own patterns!
• Warm Up: Using your warm-up time to practice with open-ended problems is a great way to model your own thinking to the whole. Modeling how to solve these problems step-by-step along with the whole class can help give reluctant participants the courage and understanding to participate and ready participants the reassurance that they’re on the right track. As an example, look at “Activity 3: Creating and Solving Problems” in the downloadable resource. You will notice a few thinking mats included, along with cards that correspond to the mats. For a warm-up activity before you begin your lesson for the day, you could give each student a laminated thinking mat and a corresponding manipulative (like, pass out the table and basket cards and give every student some small apple erasers). Then, project a corresponding task card so that everyone can see it. Read the card together, model one way you could solve the problem using your own thinking mat and manipulatives, and then allow students to solve it their own way to find one or two other answers. I would ask students to record their thinking and answers in a math journal or something similar so I could look back on their skills from early in the year and compare them to the end of the year. This is a quick, great way to collect data on student’s skills without a lot of involvement from you!

These are just a few ways to incorporate open-ended problems into your math time. I encourage you to try one way for a week or two and then experiment with another way once your students are showing they feel confident in the first implementation.

How do I make sure to provide students with open ended math problems during math each day?

In order to provide your students with activities and resources that encourage deep thinking and allow every student to participate, detailed planning is required. Deciding what standards and concept you want to focus on and choosing the best way to practice skills related to that concept before having students complete an activity is crucial to creating an effective learning time. If you peek at page 37 of the resource below, you’ll see a planning page that you can use to plan out the open-ended activities you want to use in your classroom. By editing this page with your own information, you can plan for a week of open-ended activities quickly and efficiently. This is also a great way to hold yourself accountable for how often you’re giving your students the opportunity to work on open-ended activities.

Any way you choose to implement open-ended problems in your classroom, your students are sure to grow in their problem-solving abilities and confidence. Creating a space that is safe for your students to take chances and risks with their learning is one of the greatest gifts you can give them. By incorporating ways for your students to express their individual ways of thinking, like open-ended math problems, you’ll foster a love of creative thinking and confidence in problem-solving skills.

Notes About the Included Resources:

The resources included in this blog post are for you to use in your classroom with your students. More detailed explanations for how to incorporate the activities are included in the resources themselves. There are also a few blank templates within the resource so that you can create your own task cards, thinking mats, and activity plans.

Related Professional Development Courses:

Math Stations for PreK-2nd Grade

Efficient Classroom Processes

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(10 PD Hours) Explore the strategies involved in planning and executing inquiry-based lesson plans. The components of an inquiry-led lesson will be outlined and discussed, and participants will learn to develop and plan an inquiry-led lesson.

Project Based Learning

(22 PD Hours) This course gives participants a thorough understanding of Project-Based Learning, including the critical components, as well as common misconceptions. Throughout the course, participants will design their own cycle of PBL they can immediately implement in their own classroom.

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DOWNLOADS & RESOURCES

Open Ended Math Activities for the Primary Classroom

Use these templates and graphic organizers for students who may need additional support. Feel free to download and modify the editable version, including the Frayer model template and word bank template.

IMPLEMENTATION GOAL

Choose or create an open-ended math activity to incorporate into your math instructional time. Plan to introduce the activity to your students at the beginning of the week, model and practice how to complete the activity together, and then allow them to work on the activity for 10-15 minutes per day throughout the rest of the week. Take anecdotal notes about growth you notice and how your students react to these kinds of problems. Do they enjoy them? Dread them? Are you seeing improvement in their thinking and willingness to participate? Take note of these kinds of things as the week goes on. Then, decide what open-ended problems you’ll implement the following week.

• What is Open-Ended Problem Solving? – https://mste.illinois.edu/users/aki/open_ended/WhatIsOpen-ended.html
• The Effect of Open-ended Tasks- http://journals.yu.edu.jo/jjes/Issues/2013/Vol9No3/8.pdf
• Clip art generated by Creative Clips Clipart by Krista Walden, http://www.teacherspayteachers/store/krista-walden

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Open-Ended Math Questions Reveal Student Thinking

Careful, intentional, and mindful questioning is one of the most powerful tools a skillful teacher possesses (Costa & Kallick, 2000).  Teachers can use open-ended questions during math instruction or assessments to learn how students are problem-solving.

A question is considered open-ended when it is framed in such a way that a variety of responses or approaches are possible (Small, 2009).  As shown in Figure 1, open-ended math questions are designed to uncover student understanding and misunderstandings. The responses are used to inform instruction rather than to make evaluative decisions (Rose & Arline, 2009).

Teachers analyze students’ responses to questions in order to learn how they think.  The responses reveal what students know and how they apply that knowledge.  Teachers then use this information to design instruction that supports student learning.  Additionally, open-ended questions provide opportunities for students to respond and contribute at their respective levels.  This is especially important for struggling students, since they are likely to be passive learners (Lovin, Kyger, & Allsopp, 2004).

Figure 1 illustrates how asking an open-ended question identifies students’ understanding along a continuum.  Teachers use this information to determine where to begin new instruction.

Figure 1. Continuum of student responses and instructional decisions.

The examples below from Instructional Consultation and Assessment Team (ICAT) Manual Book 3 (Gravois, Gickling, & Rosenfield, 2011, pp. 82-84) illustrates examples of open-ended questions what the responses reveal about students’ understanding of a variety of math concepts.

Specific Assessment: Mathematical Thinking

Good Math Questions Share the Following Features

• Begin with what the student knows
• Engage students in the math skills and thinking that you are trying to assess
• Require more than facts to resolve
• Are open-ended and have more than one answer

Consider the following types of mathematical open-ended questions:

Number Sense

• Can the student count 12 items?
• Does the student know how to begin?
• Does the student start with 12 and count back, or does she start with a number and add on to 12? (i.e. 10 or 5)
• Does the student need counters or can he solve it mentally?
• Does the student demonstrate knowledge of an operational algorithm?
• What factors of 6 does the student use?
• Does the student know how to begin the problem?
• Does the student use counters or can she solve the problem mentally?

[showhide type=”post2″ more_text=”Addition and Subtraction Examples… show more” less_text=”Addition and Subtraction Examples… show less”]

• How does the student approach the problem?
• Does the student understand that something is being given away?
• Can the student count to 8?
• Does the student need manipulatives?[/showhide]

[showhide type=”post3″ more_text=”Multiplication Examples… show more” less_text=”Multiplication Examples… show less”]

• Does the student add each leg?
• Did the student multiply a number or group of number by 2?

[showhide type=”post4″ more_text=”Division Examples… show more” less_text=”Division Examples… show less”]

• Does the student use flexible strategies to solve and organize the problem?
• Does the student understand division?
• Does the student recognize the different steps involved in the problem?
• Does the student draw a picture or use manipulatives?[/showhide]

[showhide type=”post5″ more_text=”Fraction Examples… show more” less_text=”Fraction Examples… show less”]

• Does the student understand ¼?
• Does the student choose a set easily divided by ¼?
• Does the student understand parts of a set as opposed to parts of a whole?[/showhide]

[showhide type=”post6″ more_text=”Probability Examples… show more” less_text=”Probability Examples… show less”]

• Does the student know what likely means?
• Does the student see the problem as a fraction, percent, or ratio?[/showhide]

[showhide type=”post7″ more_text=”Algebra Examples… show more” less_text=”Algebra Examples… show less”]

• Jacob wrote this statement on his homework paper: 8 + 6 = 7 + 3 + 4 Is this statement true?  How do you know?
• Does the student work from right to left?
• Does the student see the equal sign as meaning the same, or does he see it as an action?
• Does the student add mentally or use another strategy to add the numbers?[/showhide]

[showhide type=”post8″ more_text=”Measurement Examples… show more” less_text=”Measurement Examples… show less”]

• Does the student recognize and understand perimeter ?
• Does the student use a number of sides that represents a multiple of 36?[/showhide]

[showhide type=”post9″ more_text=”Geometry Examples… show more” less_text=”Geometry Examples… show less”]

• Does the student recognize a shape within shapes?
• Does the student recognize that the triangles keep repeating?
• Can the student explain how he found the triangles?
• Does the student understand parallel ?
• Can the student make more than two items parallel?[/showhide]

[showhide type=”post10″ more_text=”Data Analysis Examples… show more” less_text=”Data Analysis Examples… show less”]

• What type of graph does the student use to display this data?
• How does the student interpret the data on the graph?
• Does the student understand time as intervals?
• Does the student understand that the data changes over time?
• How does the student interpret the data on the graph?[/showhide]

[youtube]https://www.youtube.com/watch?v=_ofQ_WnQiZ4[/youtube] (Burn, 1993a)

[youtube]https://www.youtube.com/watch?v=1puQxclB2aw#t=10[/youtube] (Burn, 1993b)

Note to Readers: Please share how you are using open-ended questions in your classes or ask a question related to this topic by posting a comment at the end of this article.

Burn. M. (1993a).   Mathematics: Assessing, understanding Cena. Retrieved from  https://www.youtube.com/watch?v=_ofQ_WnQiZ4

Burn, M. (1993b). Mathematics: Assessing, understanding Jonathan . Retrieved from https://www.youtube.com/watch?v=1puQxclB2aw#t=10

Costa, A. L., & Kallick, B. (Eds.). (2000). Activating and engaging habits of mind. Alexandria, VA:  Association for Supervision and Curriculum Development.

Gravois, T., Gickling, E., & Rosenfield, S. (2011).   ICAT manual book 3 . Baltimore, MD: ICAT Publishing.

Lovin, A., Kyger, M., & Allsopp, D. (2004). Differentiation for special needs learners.      Teaching Children Mathematics , 11 , 158-167.

Rose, C., & Arline, C. (2009). Uncovering student thinking in mathematics, Grades 6-12. Thousand Oaks, CA: Corwin Press.

Small, M., (2009). Good questions: Great ways to differentiate mathematics i nstruction. New York, NY: Teachers College Press.

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Open-Ended Tasks and Questions in Mathematics

by CristinaM. | Sep 13, 2014 | inquiry , math , thinking | 5 comments

One way to differentiate in math class is creating open-ended tasks and questions (I talked about several differentiation strategies I use here – Mathematically Speaking ).

I think it is useful to clarify the scheme of mathematical problems – below I used Foong Pui’s research paper:

“Problems in this classification scheme have their different roles in mathematics instruction as in teaching for problem solving, teaching about problem solving, or teaching via problem solving.”

1.   CLOSED problems   are well-structured problems in terms of clearly formulated tasks where the one correct answer can always be determined in some fixed ways from the necessary data given in the problem situation.

 A. Routine closed problems – are usually multi-step challenging problems that require the use of a specific procedure to arrive to the correct, unique, answer.

B. Non-routine closed problems – imply the use of heuristics strategies * in order to determine, again, a single correct answer.

*Problem-solving heuristics: work systematically, tabulate the data, try simpler examples, look for a pattern, generalize a rule etc.

Routine problem : Minah had a bag of rice. Her family ate an equal amount of rice each day. After 3 days, she had 1/3 of the rice left. After another 7 days, she had 24 kg of rice left. How much rice was in the bag at first?

Non-routine problem : How many squares are there in a chess board?

2. OPEN –ENDED problems – are often named “ill-structured” problems as they involve a higher degree of ambiguity and may allow for several correct solutions. Real-life mathematical problems or mathematical investigations are of this type – e.g. “How much water can our school save on a period of four months?” or “Design a better gym room considering the amount of money we can spend.”

FEATURES of open-ended problems :

• There is no fixed answer (many possible answers)
• Solved in different ways and on different levels (accessible to mixed abilities)
• Empower students to make their own mathematical decisions and make room for own mathematical thinking
• Develop reasoning and communication skills

HOW do you create open-ended tasks?

Usually, in order to create open-ended questions or problems, the teacher has to work backwards :

• Indentify a mathematical topic or concept.
• Think of a closed question and write down the answer.
• Make up a new question that includes (or addresses) the answer.

STRATEGIES to convert closed problems/questions

• Turning around a question

CLOSED: What is half of 20?

OPEN: 10 is the fraction of a number. What could the fraction and the number be? Explain.

CLOSED:  Find the difference between 23 and 7.

OPEN: The difference between two numbers is 16. What might the numbers be? Explain your thinking.

CLOSED: Round this decimal to the decimal place 5.7347

OPEN: A number has been rounded to 5.8. What might the number be?

CLOSED: There are 12 apples on the table and some in a basket. In all there are 50 apples. How many apples are in the basket?

OPEN: There are some apples on the table and some in a basket. In all there are 50 apples. How many apples might be on the table? Explain your thinking.

• Asking for similarities and differences.

Choose two numbers, shapes, graphs, probabilities, measurements etc. and ask students how they are alike and how they are different.

Example: How are 95 and 100 alike? How are they different?

Possible answers:

They are alike because you can skip count by 5s, both are less than 200, both are greater than 90 etc.

They are different because one is a three-digit number, only one ends in 5, only one is greater than 99 etc.

Example: How are the numbers 6.001 and 1.006 alike? How are they different?

• Asking for explanations.

Example: Compare two fractions with different denominators. Tell how you compare them.

Example: 4 is a factor for two different numbers. What else might be true about both numbers?

• Creating a sentence

Students are asked to create a mathematical sentence that includes certain numbers and words.

Example: Create a sentence that includes numbers 3 and 4 along with the words “more” and “and”.

• 3 and 4 are more than 2
• 3 and 4 together are more than 6
• 34 and 26 are more than 34 and 20 etc.

Example: Create a question involving multiplication or division of decimals where the digits 4, 9, and 2 appear somewhere.

Example: Create a sentence involving ½  and 64 and the words “less” and “twice as much”.

• Using “soft” words.

Using the word “close” (or other equivalents) allows for a richer, more interesting mathematical discussion.

Example: You multiply two numbers and the product is almost 600. What could the numbers have been? Explain.

Example: Add two numbers whose sum is close to 750. What can the numbers be? Explain.

Example: Create two triangles with different but close areas. (*instead of, “Create a triangle with an area of 20 square inches.”)

……………………………………………………………………………………………………………………………………………………………………………………………………

A few important considerations are to be made when creating open-ended problems or questions.

• Know your mathematical focus .
• Develop questions with the right degree of ambiguity (vague enough to be interesting and to allow for different responses, but not too vague so as students get frustrated).
• Plan for two types of prompts :
• enabling prompts (for students who seem unable to start working)
• extension prompts (for students who finish quickly)

High quality responses from students have the following features:

• Are systematic (e.g. may record responses in a table or pattern).
• If the solutions are finite, all solutions are found.
• If patterns can be found, then they are evident in the response.
• Where a student has challenged themselves and shown complex examples which satisfy the constraints.
• Make connections to other content areas.

……………………………………………………………………………………………………………………………………………………………………………………………………………….

References:

Designing Quality Open-Ended Tasks in Mathematics , Louise Hodgson, 2012

Using Short Open-ended Mathematics Questions to Promote Thinking and Understanding , Foong Pui Yee, National Institute of Education, Singapore

Good Questions – Great Ways to Differentiate Mathematics Instruction , Marian Small, 2012

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Thank you for posting this. I appreciate how there is a comparison between the two (closed and open ended) types of questions and the considerations that go along with each. Thanks!

You are welcome!

Wow, well-written, thank you. I’m excited that my teaching is getting great, clear, and organized at the level that I’m at. But this article reminds me there are many higher levels I can get to, including this area of more open-endedness. Thank you!

I am happy to have helped even in a small way!

Thank you so much

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Open-Ended Maths Problems Task Cards

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Are you looking to extend your students’ mathematical thinking? Are you seeking ways to apply maths concepts you have taught engagingly and authentically? Are you after easy yet effective ways to differentiate and assess mathematical understandings? If so, these open-ended maths task cards are the perfect resource for your class.  

With 24 open-ended problems across a range of mathematics content areas, this set of cards challenges students to flexibly combine and apply a range of mathematical skills, with the opportunity to demonstrate multiple answers.

These task cards can be used for independent or guided practice. Selected cards can be used as formative assessment tools within focused units, or the set can be used throughout the year to continue to consolidate prior learning, regardless of your classroom’s current topic.

Designed with four problems per page and with the option of colour or black and white cards, simply print, cut and distribute to students. (They can also be laminated to be used again and again with different students).

Ensure your students get the most out of these challenges and know how to show their best thinking with our Solving Open-Ended Problems PowerPoint. Or you can have all the open-ended challenges in one place in our Open-Ended Maths Problems Booklet.

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April 27, 2012 / 7 Comments

Open-ended problems in elementary school mathematics

I’m hoping to find (or potentially build, given how well my search is going) some open-ended problems appropriate for elementary school math classes. By open-ended problems, I mean problems which:

• do not have an obvious solution,
• require some time to figure out,
• have multiple solutions,
• may require some assumptions are made by the students,
• are extendable in some way,
• require that the solution be explained, rather than a single number given as the answer.

I’ve found that the definition of open-ended problem seems to vary quite a bit, with many sources that I’ve found using free-response or open-response as a synonym for open-ended.

Here’s a sample question (forgive the wording, it may need improvement).

Ellen is planning a party for her friends. She has invited 100 of them, but she doesn’t know exactly how many of her friends will attend. She wants to put out tables for her friends, and she wants to put enough chairs at each table so that none of her friends has to sit alone. Assume that her friends will fill up each table as they arrive. How many tables should she put out, with how many chairs at each table?

The curriculum link here is either counting (likely to be a slow technique so I’d recommend reducing the number of friends if this is the strategy your students are going to use), addition, multiplication, or division. Note that if you do questions like this, it is important for students to explain their reasoning, and you may need to help some students do this. You may also have to point out that since Ellen doesn’t know exactly how many of her friends will attend, this problem is harder than it looks. Also, I may or may not give the actual diagram as this likely gives away too much of the problem to students. Once students have drawn a diagram though, one could turn this into a bit of a probability question (given the diagram above, how likely is it that one of Ellen’s friends will have to sit alone?).

Does anyone else know a source of questions which are this open-ended, and are designed for elementary school students? 

Here are some resources I’ve been given or found so far:

• Numeracy tasks organized by Peter Liljedahl
• Good Questions: Great Ways to Differentiate Mathematics Instruction by Marian Small
• More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction by Marion Small
• Creative Problem Solving in School Mathematics by George Lenchner
• Galileo Math problems
• Nrich Maths
• Open ended questions in elementary school math  by Mary Kay Dyer and Christine Moynihan
• Open ended assessment tasks
• Author info

Add yours →

Matt Fletcher says:

This is exactly what we’re looking to do for our Grade 6 Math program next year. We’re trying to ditch our text and our ‘traditional’ way of teaching Math strand-by-strand, and instead focus on problem solving tasks involving multiple strands at one time and require the use of the Math Processes as outlined in the Ontario Curriculum.

We’ve found the same issue you have – that being there really isn’t a great resource out there. So far, it has involved a lot of searching on the internet and in some text books, and adapting problems which may come from a Secondary School, for our grade level.

One resource I have found useful is Math that Matters by David Stocker. While not always open-ended, the questions are based on Social Justice issues and are able to generate an incredible amount of discussion surrounding student answers.

April 30, 2012 — 8:09 pm

Gary Tupper says:

Having never taught elem, & being retired now for 12 yrs, the following may be a non-starter, but:

Little Johnnie is going to the store to buy a pair of Reeboks? which he knows will cost between $65 and $70. How much money (which denominations & coinage) should he take in order to pay the exact amount? Ie the least that will be sufficient.

Extension: The following week little Johnnie and his friend Willis are heading off to the mall to buy some game cartridges. Johnnies from Walmart will be between $20 and $25; Willis’ from Zellers will be between $35 and $40. If they pool their money, what is the least amount they need in order to pay for both exactly, without change.

April 30, 2012 — 8:41 pm

David Wees says:

I think those are both good examples of open-ended problems which fit into the curriculum nicely. I believe that in grade 1 they start looking at adding coin values together, and in grade 2 they continue that process. Thank you Gary.

May 2, 2012 — 11:47 am

Kristen Beck says:

Here are some resources I love using with my kiddos:

Just for the Fun of It! Book 1 and Book 2 – AIMS Education Foundation In the Balance Algebra Logic Puzzles – Creative Publications Get It Together Math Problems for Groups – EQUAL

I’m not sure if you can find this one but it is from Oregon Council of Teachers of Mathematics: OCTM Intermediate Problem Box Problems from TOMT, 1980 to 1992 Here is the link to their publications: http://bit.ly/Ivrrvq

I use these resources with my kiddos.

May 3, 2012 — 6:02 pm

Jerry says:

Thanks for sharing! I am a fan of open ended math questions for student to work on, because not only are they searching to solve the problem through critical thinking, but they are also creatively thinking in their own way, and I believe this type of example fits in well to a type of math examples students are personally engaged in as they are trying to find their own solution to the problem. Open ended math examples are great! Thank you for sharing!

October 4, 2016 — 2:59 am

Moriah says:

Thanks for sharing! I love open ended math questions as they give learners a chance to explore concepts for themselves and come to their own understandings and conclusions. As a tutor I find critical thinking to be one of the biggest skills kids lack across the board; this is a great task to help develop these skills.

February 9, 2017 — 2:58 am

Thanks for the post! The thing i love about open-ended math questions is that it really promotes the exploration of critical thinking, as well as creativity. These concepts are so important for the intellectual development of students, and help with all subjects and ways of life; not just math. This is something that I believe needs to be focused on more, as I agree with what Moriah said about students seeming to really lack/struggle with critical thinking.

February 23, 2017 — 12:25 am

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Open-ended Maths Problem Solving Cards - Lower Primary

Updated:  26 Mar 2018

A set of 20 open-ended problem solving cards covering a range of mathematical concepts.

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Pages:  20 Pages

Years:  F - 2

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These open-ended problem solving cards will promote deep, thoughtful and creative responses from your students. More than one answer is acceptable; exploring possibilities is encouraged. The problems cover a range of mathematical concepts, such as number, space, measurement, chance and data.

Use these problem solving cards at the beginning of your lesson, as a group activity or as an enrichment task for early finishers.

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Write a review to help other teachers and parents like yourself. If you'd like to request a change to this resource, or report an error, select the corresponding tab above.

These are great, thanks. Many 'problems' people create are not true tasks that require the thinking I want from my students. With these, they must explain and justify their thinking. That is what I was seeking. I have created open ended questions of my own too, but at this price, it was great value to me as you have saved me time!

Thank you so much for taking the time to share this lovely feedback with us, Lin. And yes, it's so important for our students to learn how to justify and explain their thought processes when solving open-ended questions such as these. I'm so pleased that we saved you some time, also! Have a wonderful day.

Hi there This set of open ended math tasks were great for my Grade 1/2 class. They loved them and eagerly finished one to begin another. hHghly recommend. Tania

Hi Tania, Thank you for your positive feedback. It is great to hear that your students are loving the open-ended tasks, feel free to share some examples of their work! Kind regards, Victoria.

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The development of elementary teachers’ personal meanings of problem-solving through engaging in open-ended tasks

• Original Article
• Published: 01 July 2024

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• Serife Sevinc   ORCID: orcid.org/0000-0002-4561-9742 1 ,
• Dionne Cross Francis 2 ,
• Rick Hudson 3 &
• Jinqing Liu 4  

In this study, we explored elementary school teachers’ experiences working on open-ended mathematics tasks during a 10-day professional development (PD) workshop. Teachers engaged with the tasks daily in a session call Morning Math (MM). Thirty-two elementary teachers from three school districts in the USA participated in a 2-year professional development (PD) program. Analysis of videos of teacher engagement in the tasks and interviews shows that the use of open-ended mathematics tasks embedded within problem-solving focused PD could shape elementary school teachers’ personal meanings about learning mathematics through problem-solving. The findings indicate that the teachers’ take-aways from the PD were not limited to the mathematical content of the MM problems (i.e., mathematical residue), instead teachers’ experiences with open-ended mathematics tasks engendered personal meanings reflecting pedagogical residue with cognitive, affective, and social components. This result shows the crucial role of PD in shaping teachers’ mathematics-related personal meanings which research suggests that it will subsequently influence their beliefs and instruction.

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Teachers worked on MM problems collaboratively in self-selected groups; that is, the number of years of teaching experience was not considered as a criterion to make homogenous or heterogeneous groups.

All teacher names used in this manuscript are pseudonyms, and the numbers given in brackets as subscripts indicate the year of teaching experience.

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Will Biden Withdraw?

Democratic worries about the president’s age have surged after thursday’s debate..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From “The New York Times,” I’m Natalie Kitroeff. This is “The Daily.”

[MUSIC PLAYING]

President Joe Biden’s disastrous debate performance last week set off a furious discussion among Democratic officials, donors, and strategists about whether and how to replace him as their party’s nominee. Today, chief White House correspondent Peter Baker takes us inside those discussions and Biden’s effort to shut that conversation down.

It’s Monday, July 1.

Peter, you’ve been reporting on what I think can be best described as the great Democratic freakout that started basically from the moment the debate began at 9:00 PM on Thursday night. Tell us about the aftermath.

Yeah, I’ve been covering politics for 38 years, and I’ve never seen a political panic like we saw after that debate. It was like a run on the bank. Everybody in the Democratic Party was suddenly confronted with what they didn’t want to admit up until then, which is that they have an 81-year-old candidate who would be 86 at the end of his second term. And it’s very possible that he was not capable of completing this campaign in a vigorous and competitive way against Donald Trump. That’s what really it comes down to for many Democrats. Can Joe Biden take the campaign to Donald Trump and stop what they think is an existential threat to the country?

I want to know more about who you were hearing from. Who are the people that are calling you? What are the big questions they’re asking? What are they struggling with?

Yeah, I don’t want to get into too many names. A lot of people don’t want to be out front. But you did see even publicly, people like Senator Claire McCaskill.

Joe Biden had one thing he had to do tonight and he didn’t do it.

The former Senator from Missouri, red state Democrat, was on MSNBC just minutes after the debate.

He had one thing he had to accomplish, and that was reassure America that he was up to the job at his age. And he failed at that tonight.

She talked about this was a crisis, that her phone was blowing up with a lot of Democrats. And she was very forthright about it. It was very striking that she said that.

I think there’s a lot of people who are going to want to see him consider taking a different course now,

People like van Jones, who was on CNN, he used to work in the Obama White House.

We’re still far from our convention. And there is time for this party to figure out a different way forward, if you will allow us to do that.

He very candidly talked about how this was going to raise questions about whether the President should continue as the candidate.

Some Democrats are calling for Biden to step down. Andrew Yang —

Andrew Yang, who ran against Biden in 2020 for the Democratic nomination, popular with some younger voters, he said on social media it was time for Biden to step aside. Those are some of the public people. And obviously, in the hours and days that followed, more came out and said, well, this is something we need to think about.

But the people I was talking to were people behind the scenes, people who have run White Houses before, people who work for President Biden in this administration. I heard words like, “He can’t win.” “This is a disaster.”

“This is a nightmare.” And they were very, very concerned that he could not beat Donald Trump.

Right. And you saw these really prominent media figures, outlets, “The Times” as an actor in this situation calling for Biden to step aside. Our editorial board did this. We should say this is entirely separate from our newsroom from the show, but there was this real crescendo. And there was a sense that this was a turning point, right?

Absolutely. But it’s not just the media. I think what the Biden campaign would like it to be is about the media. It’s just that the media tends to be more out front and say things more openly than Democrats were saying. It really was rank and file Democrats. It really was high ranking Democrats, and they were absolutely flipped out.

Right. These doubts are coming from all over, from many corners. Take me through, Peter, the argument for why this poor performance meant that Biden should be replaced. How do they explain that thinking?

Well, look, a lot of people who defended President Biden will say is that incumbent presidents don’t do well in their first debate, and that is true. Historically, that’s been true. Ronald Reagan, George W. Bush, Barack Obama, Donald Trump all lost, arguably, their first debate when they were running for re-election.

But the difference is, if Obama doesn’t register a good performance against Mitt Romney, first of all, nobody thought that Obama wasn’t capable of being president as a result. And second of all, he had another debate about a week or two later in order to try to recover. Neither of those factors works here.

Biden’s problem from this debate is much more existential. It’s much more profound because it’s about whether he is able to perform the office of president, not just for the next few months, but for the next 4 and 1/2 years. And there’s not going to be another debate until September. So he doesn’t have another big audience opportunity to change people’s minds, to show that, in fact, he does still have it and can run the country. And that’s a real problem for him.

And there’s this broader context here, right. Voters have been telling pollsters for a year now that Biden’s age is a major concern for them. We’ve seen Biden’s age before our very eyes. We’ve seen him stumble in speeches, in public appearances. And, Peter, we had talked to you about this very issue a few months ago after a special counsel investigating Biden’s son, Hunter, issued this report focusing on Biden’s mental state, in part, saying that the president was, quote, a “well-meaning elderly man with a poor memory” and had, quote, “diminished faculties in advancing age.”

But at the time, the White House dismissed that report as a partisan hit job. So in a sense, this debate performance was the capstone of something that’s been in the air for a very long time. It’s just that this time, it was undeniable. There was no spinning it.

Well, I think that’s exactly right. There was no spinning it. One Democrat put it to me. He said, for a long time, the fear of Trump stifled Democratic criticism of Biden. People didn’t want to criticize him because they desperately want to beat Trump.

But now, that same fear, he said to me, now meant that they could no longer stand behind Biden, that they worried that he had been diminishing over a period of time and that his staff and the people around him had hidden that from the public. There’s a real anger out there among some Democrats. Now, what the Biden circle would tell you is, no, we didn’t hide anything from you.

Yes, he does have moments where he is not as lucid as you would want him to be, but that, broadly speaking, when they see him operate, when they’re sitting with him in the Oval Office or in the situation room, he is sharp. He asks good questions. He understands and grasps the issues that he is confronting.

And I mean, we all have good days and bad days. But when you’re 81, your good days and bad days may be more pronounced.

And if he has good days and bad days, well, Thursday night was a very bad night.

Peter, I want to ask you about that, about your view on all of this, because I do think all this has raised this fundamental question for a lot of Democrats, for a lot of journalists, for voters, which is what you’re getting at. Was what we saw on the debate stage the real Biden? And had the White House been hiding him from us? Or were the people around him just unable to recognize the perils of this themselves? Like, have they been gaslighting us all, or are they in denial?

Yeah, it’s a good question. That’s the question in some ways, right. I think that people who work closely with the president and like him, admire him, respect him want to see the best in him and want everybody else to see the best in him. And they have been unwilling to admit whether or not he has slipped in the last 3 and 1/2 years.

And part of it may be strategic. They recognize in their view that he is the president. They’ve got to build him u and make him as successful as possible. And they have shielded him as much as possible from public scrutiny.

He hasn’t give as many interviews or as many press conferences as any of his predecessors going back to Reagan. He’s never getting interview to “The New York Times,” or “The Washington Post,” or “The Wall Street Journal,” or the “LA Times,” or any other newspaper, which is, I think, the first president, certainly in my lifetime, who hasn’t done that. And that’s been part of a pattern of them trying to protect him. And I think there’s kind of a reckoning right now among other Democrats, wondering whether they went too far.

OK. So we have this huge reaction to this moment, the debate from the Democrats in the news media. Can you walk us through how the Biden team responds to the full-blown panic?

His campaign was thrown into full-blown damage control over the weekend. And the President himself set out to do two things. First, privately, he met with donors and assured them, yes, he’s still a viable candidate and that they should still support him.

And then publicly, he went on a campaign blitz, traveling to seven events in four states. And his first stop on Friday, in fact, was at a rally in Raleigh, North Carolina.

[CROWD CHEERING]

Hello, hello, hello.

This was already scheduled before the debate, but it gave him an opportunity to both show that he can do the job —

Thank you, North Carolina.

— to demonstrate vigor and vitality —

I don’t know what you did last night, but I spent 90 days and 90 minutes on the stage debating the guy who has the morals of an alley cat.

— and to address his own performance.

I know I’m not a young man. State the obvious. Well, I know.

And he says pretty candidly, he says, yeah, I’m not a young man.

I don’t walk as easy as I used to. I don’t speak as smoothly as I used to. I don’t debate as well as I used to.

But he goes on.

Well, I know what I do know. I know how to tell the truth.

And I know how to tell right from wrong.

And I know how to do this job. I know how to get things done. And I know, like millions of Americans know, when you get knocked down, you get back up.

And, you know, he comes across as pretty vigorous, pretty energetic. Of course, he’s reading from a teleprompter. Always a lot easier to read from a teleprompter. But I think more important than that was the body language and the spirit that he brought to the moment.

It sounds like teleprompter, not the Biden who appears at the rally is meaningfully different, at least in style from the President that we saw on the debate stage. I’m wondering if you think this has changed anything. Has the conversation changed?

No, not fundamentally. I think fundamentally that people still recognize that there’s an issue here. Now, there was pushback among Democrats saying, OK, take a breath. Get a hold of yourself. He’s not dropping out.

I don’t think you judge a person’s — the body of their work on one night. They don’t always go the way you want to. I have confidence in the President because he’s delivered.

And I understand that he had a raspy voice. But like I’ve told folks, who cares? We have a choice this November between someone that’s a good person, a good president with a real record of results, and someone that has brought shame on the presidency.

Don’t let 90 minutes define a career of a president who’s been in office for 3 and 1/2 years, been in politics for 50 years, and overshadow the important issues that he stands for. And so you heard that line of thinking in the spin room and on TV.

Look, I think Joe Biden had a bad debate night, but it doesn’t change the fact that Donald Trump was a bad president.

And by the way, Trump did terribly, too, which is a fair point. Trump may have been more lucid in the sense that he sounded stronger. But if you actually looked at what he said, listen to what he said, he said so many things that were just not true. And it helped Biden that former President Barack Obama put out a statement saying, hey, guys, I’ve seen bad debates. It’s fine. Don’t freak out, in effect, is what he said.

And Jim Clyburn —

And if he asked my opinion, I would give it, as I always do —

— who is his very close ally in Congress from South Carolina, the Congressman who helped get him the nomination in the first place, said, stay the course.

He should stay in this race. He should demonstrate going forward his capacity to lead the country.

So it was important to have those voices out there among prominent Democrats trying to calm the waters. But it only went so far because the waters are still churning underneath.

Peter, I’m curious how his donors are reacting to all this. I mean, you mentioned that part of his full court press is to reassure them that he’s got the mental acuity to run. How successful has he been at that?

Yeah, I think that there are certainly some donors who are resigned. They feel like there’s not much choice. But there are others who actually are considering jumping off the boat. Jumping on what Mika Brzezinski on “Morning Joe” called the hysteria train. And I think that it’s an open question.

But part of the thing is, of course, they’re waiting to see how the polls really shake out. The initial polls after any debate are often not really representative of how an event settles into the political narrative. And the polling and data so far have been kind of contradictory. On the one hand, it shows that Trump clearly beat Biden. Biden clearly lost, and that Biden has only reinforced the doubts that most voters have about his age and mental capacity. That’s absolutely true.

At the same time, there’s some polling showing that the overall horse race number, who are you going to vote for, hasn’t moved dramatically yet, if it does at all, and that it’s possible this is baked in that people who were going to vote against him are still going to vote against him. The people who would vote for him, holding their nose, may not be happy about it, may still be voting for him.

But there’s a tell. The tell was from the Biden campaign. When they put out a memo by Jen O’Malley Dillon, who was his top political person at the campaign, and she says if you see polls go down in the next few days or weeks, what’s telling is that she is, in fact, anticipating that polls would be bad for them and trying to set expectations for supporters and voters and donors saying, don’t let that panic you any further. That’s normal, and we’ll get past that just as we have other bumps in the road.

It seems like the Biden effort over the weekend has, in some sense, quieted some public doubts from key Democrats, right? There’s not a — we didn’t see a deluge of senior lawmakers going on Sunday talk shows and saying, Mr. President, step aside. But from what you’re saying and based on the reporting that we’ve seen from our colleagues, the effort has not, by any means, ended the discussion about replacing Biden. That is very much still happening under the surface.

Yeah. That discussion is very much alive among Democrats. Will Biden and should Biden remain as the candidate? And the question then becomes is if he doesn’t, what then?

We’ll be right back.

Peter, given that this discussion of Biden stepping aside is still, as you said, very much alive, what would it look like for someone to replace him on the presidential ticket at this point in the campaign just a few months before election day? It sounds like it would be pretty daunting.

Yeah. I mean, look, we have never had a situation like this, not certainly in modern times. No president has ever dropped out of the race so late in the cycle. And you have to remember a couple of things.

First of all, the Democratic National Convention, which would anoint a new nominee, is in late August. But they’re actually scheduled to take a roll call vote before the convention begins on August 7. So that means we have five weeks between now and when the roll call is scheduled to be held to decide a nominee.

If the president were to drop out, that would create this truncated, incredibly intense, incredibly wide open, incredibly volatile, short campaign to figure out who would be the nominee. And it’s complicated logistically. It’s complicated politically, it’s complicated in all sorts of ways. And we don’t really know what’s going to happen or how it would happen because we’ve never seen it before.

But it is conceivable. It is possible. The President has to decide that he’s not going to run. If that doesn’t happen, then there’s no contest. There’s no way anybody sees a forcing him off the ballot if he chooses to continue to run. That doesn’t seem to be any appetite for trying to find a way to undo his nomination other than with his consent.

He controls the 3,900 delegates that are going to be at the convention. They’re obligated to vote for him on the first ballot. So it has to be first, his decision on whether he continues to run. If he does, then that’s it. That’s the end of that question. But if he doesn’t, then it’s jump ball.

Would his replacement automatically be Kamala Harris as the vice president?

No, not at all. If it were after the convention and they were both nominated, and then he stepped aside at the last minute, then they probably would simply go to Kamala Harris because she had been ratified by the convention as the vice presidential candidate. That’s possible. But if we’re talking about a situation before the convention, it’s anybody’s guess. There’s about a dozen other prominent Democrats out there who are looking at jumping in if suddenly, the nomination is up for grabs.

But so who are we talking about? What are some of the most prominent names that have come up?

Well, other than Kamala Harris, you have a number of governors, particularly Gavin Newsom of California, Gretchen Whitmer of Michigan, J.B. Pritzker of Illinois. There are, of course, those who ran last time who might jump back in, in theory. Senator Cory Booker, Senator Amy Klobuchar, potentially even maybe Pete Buttigieg, who is currently the transportation Secretary.

But the ones who are going to have the best chance are those who have an apparatus already, who have a set of donors and fundraisers who can raise money instantly, and who have the ability to get on TV and get media attention without having to work as hard for it. And that does suggest, obviously, a sitting vice president or a sitting governor.

Basically hitting reset on the whole nomination process requires embracing a moment of genuine political chaos. I mean, that’s the theoretical downside of this. But of course, I mean, there’s also a potential upside, right? Massive media attention potentially for whoever the replacement Democrat is, the possibility that that replacement could energize a lot of Democratic voters and independents, and even potentially moderate Republicans out there who dislike Trump but just couldn’t get excited about Biden.

Yeah, absolutely. Whoever emerges will have a certain advantage of freshness, right. And that person will have a generational argument to make against Trump because whoever it would be would be younger than Trump. And suddenly, Trump is then the old candidate. He’s 78.

And that new Democratic candidate would be able to say, I’m the next generation. This guy is also too old to be president. And if you’re concerned about our guy was, can he make it through four years, then you should be picking me because the other guy can’t make it either through four years.

Now, the downside, of course, is these are people who are largely untested on a national stage, at least in this kind of an environment. And you don’t know how people will do once they actually jump in. They didn’t have the advantage of a year-long primary contest to prove themselves.

Before you jump into a race, you can look really attractive. Look at Ron DeSantis. Before he jumped in to the Republican primaries, on the Republican side, they all thought, wow, he’s really great. Didn’t turn out to be so great once he got on the campaign trail. The magic didn’t actually appear. So that’s the danger here, is we don’t know which of these Democrats, if any, would have the ability to shine when the big giant klieg lights are on them.

And we’re clearly in a delicate moment right now. But are any of these potential contenders trying to signal interest at this point? I mean, what does that look like? It sort of seems like the art of raising your hand for something but not wanting it to publicly.

Yeah, it’s a really good question because you obviously can’t do anything that seems disloyal to Biden, right. Nobody’s willing to take on Biden directly and say, I’m now running, and you should take him down. So the trick here is you have to be loyal, loyal, loyal right up to the minute that Biden says he’s not running, at which point then suddenly you’re off to the races.

And doing that from a standing start is not a good idea for any campaign. So they have to find ways of talking to their people, lining up donors, thinking about what kind of a campaign would look like, who might be their strategist, without letting anybody know that they’re doing that, or at least not let anybody in our business know that because it would obviously backlash on them. And that’s a very hard thing to do. I mean, I heard that there are people out there making phone calls who want to run, but nobody’s going to admit that outright because it would be damaging to them.

And in terms of logistics here — I know this is all very hypothetical — but if a new candidate were to become the nominee, do they get all the money Biden raised? Do they get his campaign team, or are they literally creating a presidential campaign from scratch with four months to go?

I mean, presumably, they would adopt a lot of Biden’s apparatus. As for the money, a lot of the money these days is in kind of superpacs and these sort of amorphous structures that can go immediately to a different candidate. And everybody who contributed to Biden can now contribute to the new candidate. And then Biden can still spend his money as he chooses in support of whoever the candidate is.

So there’s that advantage in a way it could actually increase some fundraising. But you’re right, they would be starting from scratch in a lot of ways, at least in terms of a national organization.

And obviously, for this to even happen, it relies on Biden stepping aside here, as you said. Everything you’ve laid out so far suggests that he is, for now at least, closed off to this suggestion. And I have to ask what you think from your reporting would change that.

Well, President Biden is a proud man. He’s a stubborn man. As a lot of people of any age are, he is reluctant to confront and face his own weaknesses, and he’s not going to be talked out of running by a bunch of media chattering class, pundits, and junior Democrats. I mean, think about it. He’s been running for president since 1987. And the idea that he is now in office and running the country and as he thinks it, running it pretty well, that he’s going to simply step aside because a bunch of people tell him he should, he reacts viscerally to that. Of course, he doesn’t want to do that.

In fact, when you talk to Democrats, they’re very conscious of not trying to push him because it could have the opposite reaction. It could trigger him to want to stay even more. The people who have influence with him, not that many people.

I mean, at this point, he’s been in politics since 1972 when he was elected to the Senate. And the people he considers his peers, most of them are gone. He’s not surrounded by people whose opinion he truly respects.

Obama and Clinton, the only two former presidents out there other than Jimmy Carter, who are Democrats, I don’t know that if they told him it was time to pull the plug, that he would listen to that. In fact, he might, again, do the opposite. He still resents Obama for discouraging from running in 2016.

Obviously, congressional leaders like Chuck Schumer, Nancy Pelosi, Hakeem Jeffries, Jim Clyburn, people like that, could have an influence if they were to go as a group to him in a way like the Republicans went to Nixon in 1974 and said that he wasn’t going to survive. Maybe that might influence him. But I don’t think they’re likely to do it. It doesn’t look like they’re likely to do that.

Who does have his ear? I mean, who does he really listen to? Who do we know he’s going to be listening to in this moment?

Well, the real people he listens to the most are his family. He’s a very family-oriented guy. Remember, he went home to Wilmington almost every night when he was a Senator. Even now as president, he flies home to Wilmington most every weekend.

And he’s at Camp David this weekend with his family. They’re there because they had already planned to be together for a photo shoot, ironically, with Annie Leibovitz. So the family was already gathering at Camp David —

— gives him an opportunity. Right. And it gives him an opportunity to have a heart-to-heart conversation with the people who mean the most to him, in particular, Jill Biden, of course, the first lady and his sister, Valerie, the kids, and the grandkids. We don’t know what that conversation looks like.

I mean, that’s as private as it comes. Someday they’ll be histories written and memoirs written. And I’m fascinated to read what’s happening this weekend at Camp David, because I think it’s kind of pivotal. But what we do know is that up until now, at least, Jill has been all in.

She’s been encouraging him to run. And it sounds like she’s been encouraging him to stay in. And a lot of people think her voice is the most important voice in this.

And just to pause on this for a second, Peter, it sounds like you’re saying something pretty remarkable, actually, which is that this very small handful of people have influence over what is going to be a massive decision that affects hundreds of millions of Americans and really the entire world.

Well, don’t underplay it. But yeah, that’s exactly right. And you’re right to point out the stakes here, because it’s not just about Joe Biden and his future. It is about this presidential race. And it is about whether you want Donald Trump back in office. That’s the way most Democrats look at it. And the difference between a Biden presidency and a Trump presidency is about as stark as you can imagine. And this will have consequences that will ripple out for years to come.

Peter, I kind of want to end where we began this conversation, which is with this question of Democrats’ mindset right now, post-debate, and the question of what’s motivating them in this moment. Privately, they’re talking about replacing Biden. They seem too scared to do that publicly for all the reasons we’ve discussed, including that the White House is basically telling them to stop talking about it and that Biden probably won’t step aside anyway. So all of this may just be useless hand-wringing.

And all of that raises a pretty essential question for me, which is, are Democrats about to recommit to a damaged nominee who had a disastrous debate, who may only get worse over time out of loyalty to President Biden? Or do you think on some level they really believe this is all just overblown, that it’s one debate, everybody has bad days, and Biden really can recover from all this?

I think there’s a lot of doubt in the party that he can really recover from. This is the argument obviously the Biden campaign is making. And it’s really the only argument they have available to them to say, look, don’t overreact here. It’s just the media hyperventilating. We will recover like we’ve recovered before.

And it is what they have to say because they don’t have anything else that they can say. But almost every Democrat outside of the inner circle that I talk to says this was a complete disaster. They are not sugarcoating it. They saw what they saw with their own eyes, and they think that it’s not tenable for the campaign to pretend otherwise.

And it’s particularly devastating because the weakness of the Biden campaign has always been concerns about his age. People have expressed that to pollsters and in interviews with reporters going back more than a year. And now they just saw it for themselves on TV, in their living rooms, more than 50 million of them. And to convince them not to believe what they saw with their own eyes is just a monumental task politically.

So for Biden’s team and for the Democrats, it’s a big roll of the dice. Do you stick with him and try to correct the damage in the four months you have? Or do you say, it’s time for somebody else, which is going to be a roll of the dice itself?

And assuming Biden continues, Peter, it sounds like we won’t know if that bet, that bet that he’s the only one that ever beat him. He’s the only guy that can get this done, if that was solid or diluted thinking until November 5.

On November 5, or whatever day of the election is resolved, we will know how history judges this. And maybe everybody looks back on this and says, well, that was close, but they were right to stick it out because they won. Or in fact, it was a disastrous decision, and they wasted an opportunity to fix campaign that was already having trouble. Hindsight will be perfect, but there’s no question that this weekend will be remembered as perhaps the most decisive of this campaign.

Peter, thanks for coming on.

Thanks for having me.

After we spoke with Peter, “The New York Times” reported that while at Camp David, Biden’s family urged the president to stay in the race, arguing that, yes, he could still show voters he’s capable of serving another four years.

Here’s what else you need to know today. Early projections show that France’s far-right national rally party won a decisive victory in the first round of voting for the country’s national assembly on Sunday. National rally, an anti-immigrant party long on the fringes of the French political scene, captured about a third of the vote, according to polls, and now appears poised to become the largest force in the lower house of parliament.

The results dealt a blow to French president Emmanuel Macron, who took a gamble by dissolving parliament last month and calling for snap elections. But his bet that the far right wouldn’t repeat its recent success in European parliament elections backfired. A run-off election between the leading parties will be held on July 7.

Today’s episode was produced by Asthaa Chaturvedi, Rachelle Bonja, Will Reid, and Rob Szypko, with help from Olivia Natt and Lynsea Garrison. It was edited by Lexie Diao, with help from Ben Calhoun, Paige Cowett, and Mike Benoist. And special thanks to Michael Barbaro.

Contains original music by Dan Powell, Marion Lozano, and was engineered by Chris Wood. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Natalie Kitroeff. See you tomorrow.

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Produced by Rachelle Bonja ,  Will Reid and Rob Szypko

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Original music by Dan Powell and Marion Lozano

Engineered by Chris Wood

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President Biden’s disastrous debate performance last week set off a furious discussion among Democratic officials, donors and strategists about whether and how to replace him as the party’s nominee.

Peter Baker, who is the chief White House correspondent for The Times, takes us inside those discussions and Biden’s effort to shut them down.

On today’s episode

Peter Baker , the chief White House correspondent for The New York Times.

Background reading

President Biden’s allies can no longer wave away concerns about his capacity after his unsteady performance at Thursday’s debate.

Mr. Biden’s family is urging him to keep fighting .

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

Special thanks to Michael Barbaro .

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Maddy Masiello, Isabella Anderson, Nina Lassam and Nick Pitman.

Natalie Kitroeff is the Mexico City bureau chief for The Times, leading coverage of Mexico, Central America and the Caribbean. More about Natalie Kitroeff

Peter Baker is the chief White House correspondent for The Times. He has covered the last five presidents and sometimes writes analytical pieces that place presidents and their administrations in a larger context and historical framework. More about Peter Baker

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